The sum of the first powers (i.e. of the numbers themselves):
$$\sum_{i=0}^n f_i = f_{n+2}-1$$
And the sum of the squares:
$$\sum_{i=0}^n f_i^2 = f_n f_{n+1}$$

The simplicity of these expressions suggests that similar closed form solutions exist for other powers too, and it is indeed so. Solving the above-mentioned problem, I wrote a Python3 + Sympy program, which can derive such expressions for arbitrary large powers (well, there are limits of course: computer memory and your time, but powers around 30 can be resolved in reasonable time). It seems that these formulas are not widely represented in the Internet, so I decided to share them. Who knows, maybe I'll save somebody's 5 minutes?

Source code

The repository contains several programs, main of them is fibsums_general.py, which is the script for deriving a formula. For any positive power, it gives two closed-form representations: one expresses the sum via the products and powers of \(f_n, f_{n+1}\), and another - via \(f_{kn}\). Usually, the second one is simpler.